\(\int (e+f x)^2 (a+b \coth ^{-1}(c+d x))^2 \, dx\) [109]

Optimal result

Integrand size = 20, antiderivative size = 374 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \text {arctanh}(c+d x)}{3 d^3}-\frac {2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{3 d^3} \]

[Out]

1/3*b^2*f^2*x/d^2+2*a*b*f*(-c*f+d*e)*x/d^2+2*b^2*f*(-c*f+d*e)*(d*x+c)*arccoth(d*x+c)/d^3+1/3*b*f^2*(d*x+c)^2*(
a+b*arccoth(d*x+c))/d^3-1/3*(-c*f+d*e)*(d^2*e^2-2*c*d*e*f+(c^2+3)*f^2)*(a+b*arccoth(d*x+c))^2/d^3/f+1/3*(3*d^2
*e^2-6*c*d*e*f+(3*c^2+1)*f^2)*(a+b*arccoth(d*x+c))^2/d^3+1/3*(f*x+e)^3*(a+b*arccoth(d*x+c))^2/f-1/3*b^2*f^2*ar
ctanh(d*x+c)/d^3-2/3*b*(3*d^2*e^2-6*c*d*e*f+(3*c^2+1)*f^2)*(a+b*arccoth(d*x+c))*ln(2/(-d*x-c+1))/d^3+b^2*f*(-c
*f+d*e)*ln(1-(d*x+c)^2)/d^3-1/3*b^2*(3*d^2*e^2-6*c*d*e*f+(3*c^2+1)*f^2)*polylog(2,(-d*x-c-1)/(-d*x-c+1))/d^3

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {6247, 6066, 6022, 266, 6038, 327, 212, 6196, 6096, 6132, 6056, 2449, 2352} \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=-\frac {(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {2 b \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac {2 a b f x (d e-c f)}{d^2}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \text {arctanh}(c+d x)}{3 d^3}-\frac {b^2 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}+\frac {2 b^2 f (c+d x) (d e-c f) \coth ^{-1}(c+d x)}{d^3}+\frac {b^2 f^2 x}{3 d^2} \]

[In]

Int[(e + f*x)^2*(a + b*ArcCoth[c + d*x])^2,x]

[Out]

(b^2*f^2*x)/(3*d^2) + (2*a*b*f*(d*e - c*f)*x)/d^2 + (2*b^2*f*(d*e - c*f)*(c + d*x)*ArcCoth[c + d*x])/d^3 + (b*
f^2*(c + d*x)^2*(a + b*ArcCoth[c + d*x]))/(3*d^3) - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (3 + c^2)*f^2)*(a + b*
ArcCoth[c + d*x])^2)/(3*d^3*f) + ((3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c + d*x])^2)/(3*d^3
) + ((e + f*x)^3*(a + b*ArcCoth[c + d*x])^2)/(3*f) - (b^2*f^2*ArcTanh[c + d*x])/(3*d^3) - (2*b*(3*d^2*e^2 - 6*
c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c + d*x])*Log[2/(1 - c - d*x)])/(3*d^3) + (b^2*f*(d*e - c*f)*Log[1 -
(c + d*x)^2])/d^3 - (b^2*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))]
)/(3*d^3)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 6022

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x^n])^p, x] - Dist[b
*c*n*p, Int[x^n*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p
, 0] && (EqQ[n, 1] || EqQ[p, 1])

Rule 6038

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCoth[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))
), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1
]

Rule 6056

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcCoth[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^
2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]

Rule 6066

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((
a + b*ArcCoth[c*x])^p/(e*(q + 1))), x] - Dist[b*c*(p/(e*(q + 1))), Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^(p
- 1), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] &
& NeQ[q, -1]

Rule 6096

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]

Rule 6132

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 6196

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :>
Int[ExpandIntegrand[(a + b*ArcCoth[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x]
&& IGtQ[p, 0] && EqQ[c^2*d + e, 0] && IGtQ[m, 0]

Rule 6247

Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &
& IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^2 \left (a+b \coth ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {(2 b) \text {Subst}\left (\int \left (-\frac {3 f^2 (d e-c f) \left (a+b \coth ^{-1}(x)\right )}{d^3}-\frac {f^3 x \left (a+b \coth ^{-1}(x)\right )}{d^3}+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \coth ^{-1}(x)\right )}{d^3 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{3 f} \\ & = \frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {(2 b) \text {Subst}\left (\int \frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d^3 f}+\frac {\left (2 b f^2\right ) \text {Subst}\left (\int x \left (a+b \coth ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d^3}+\frac {(2 b f (d e-c f)) \text {Subst}\left (\int \left (a+b \coth ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^3} \\ & = \frac {2 a b f (d e-c f) x}{d^2}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {(2 b) \text {Subst}\left (\int \left (\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(x)\right )}{1-x^2}+\frac {f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x \left (a+b \coth ^{-1}(x)\right )}{1-x^2}\right ) \, dx,x,c+d x\right )}{3 d^3 f}-\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}+\frac {\left (2 b^2 f (d e-c f)\right ) \text {Subst}\left (\int \coth ^{-1}(x) \, dx,x,c+d x\right )}{d^3} \\ & = \frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}-\frac {\left (2 b^2 f (d e-c f)\right ) \text {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (2 b (d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {a+b \coth ^{-1}(x)}{1-x^2} \, dx,x,c+d x\right )}{3 d^3 f}-\frac {\left (2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {x \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d^3} \\ & = \frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \text {arctanh}(c+d x)}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac {\left (2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {a+b \coth ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{3 d^3} \\ & = \frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \text {arctanh}(c+d x)}{3 d^3}-\frac {2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}+\frac {\left (2 b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d^3} \\ & = \frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \text {arctanh}(c+d x)}{3 d^3}-\frac {2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac {\left (2 b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c-d x}\right )}{3 d^3} \\ & = \frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \text {arctanh}(c+d x)}{3 d^3}-\frac {2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{3 d^3} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1115\) vs. \(2(374)=748\).

Time = 7.06 (sec) , antiderivative size = 1115, normalized size of antiderivative = 2.98 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=a^2 e^2 x+a^2 e f x^2+\frac {1}{3} a^2 f^2 x^3+\frac {1}{3} a b \left (2 x \left (3 e^2+3 e f x+f^2 x^2\right ) \coth ^{-1}(c+d x)+\frac {d f x (6 d e-4 c f+d f x)-(-1+c) \left (3 d^2 e^2-3 (-1+c) d e f+(-1+c)^2 f^2\right ) \log (1-c-d x)+(1+c) \left (3 d^2 e^2-3 (1+c) d e f+(1+c)^2 f^2\right ) \log (1+c+d x)}{d^3}\right )-\frac {2 b^2 e f \left (1-(c+d x)^2\right ) \left (\frac {(c+d x) \coth ^{-1}(c+d x)}{d^2}-\frac {c (c+d x) \coth ^{-1}(c+d x)^2}{d^2}+\frac {(c+d x)^2 \left (1-\frac {1}{(c+d x)^2}\right ) \coth ^{-1}(c+d x)^2}{2 d^2}-\frac {\log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )}{d^2}+\frac {2 c \left (\frac {1}{2} \coth ^{-1}(c+d x)^2+\coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )}{d^2}\right )}{(c+d x)^2 \left (1-\frac {1}{(c+d x)^2}\right )}+\frac {b^2 e^2 \left (1-(c+d x)^2\right ) \left (\coth ^{-1}(c+d x) \left (\coth ^{-1}(c+d x)-(c+d x) \coth ^{-1}(c+d x)+2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )}{d (c+d x)^2 \left (1-\frac {1}{(c+d x)^2}\right )}-\frac {b^2 f^2 (c+d x) \sqrt {1-\frac {1}{(c+d x)^2}} \left (1-(c+d x)^2\right ) \left (\frac {4 \coth ^{-1}(c+d x)}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {3 \coth ^{-1}(c+d x)^2}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}-\frac {12 c \coth ^{-1}(c+d x)^2}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {9 c^2 \coth ^{-1}(c+d x)^2}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {-1+6 c \coth ^{-1}(c+d x)+3 \coth ^{-1}(c+d x)^2-3 c^2 \coth ^{-1}(c+d x)^2}{\sqrt {1-\frac {1}{(c+d x)^2}}}+\cosh \left (3 \coth ^{-1}(c+d x)\right )-6 c \coth ^{-1}(c+d x) \cosh \left (3 \coth ^{-1}(c+d x)\right )+\coth ^{-1}(c+d x)^2 \cosh \left (3 \coth ^{-1}(c+d x)\right )+3 c^2 \coth ^{-1}(c+d x)^2 \cosh \left (3 \coth ^{-1}(c+d x)\right )+\frac {6 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {18 c^2 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}-\frac {18 c \log \left (\frac {1}{c+d x}\right )}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}-\frac {18 c \log \left (\frac {1}{\sqrt {1-\frac {1}{(c+d x)^2}}}\right )}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {4 \left (1+3 c^2\right ) \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )}{(c+d x)^3 \left (1-\frac {1}{(c+d x)^2}\right )^{3/2}}-\coth ^{-1}(c+d x)^2 \sinh \left (3 \coth ^{-1}(c+d x)\right )-3 c^2 \coth ^{-1}(c+d x)^2 \sinh \left (3 \coth ^{-1}(c+d x)\right )-2 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right ) \sinh \left (3 \coth ^{-1}(c+d x)\right )-6 c^2 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right ) \sinh \left (3 \coth ^{-1}(c+d x)\right )+6 c \log \left (\frac {1}{c+d x}\right ) \sinh \left (3 \coth ^{-1}(c+d x)\right )+6 c \log \left (\frac {1}{\sqrt {1-\frac {1}{(c+d x)^2}}}\right ) \sinh \left (3 \coth ^{-1}(c+d x)\right )\right )}{12 d^3} \]

[In]

Integrate[(e + f*x)^2*(a + b*ArcCoth[c + d*x])^2,x]

[Out]

a^2*e^2*x + a^2*e*f*x^2 + (a^2*f^2*x^3)/3 + (a*b*(2*x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCoth[c + d*x] + (d*f*x*(6
*d*e - 4*c*f + d*f*x) - (-1 + c)*(3*d^2*e^2 - 3*(-1 + c)*d*e*f + (-1 + c)^2*f^2)*Log[1 - c - d*x] + (1 + c)*(3
*d^2*e^2 - 3*(1 + c)*d*e*f + (1 + c)^2*f^2)*Log[1 + c + d*x])/d^3))/3 - (2*b^2*e*f*(1 - (c + d*x)^2)*(((c + d*
x)*ArcCoth[c + d*x])/d^2 - (c*(c + d*x)*ArcCoth[c + d*x]^2)/d^2 + ((c + d*x)^2*(1 - (c + d*x)^(-2))*ArcCoth[c
+ d*x]^2)/(2*d^2) - Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])]/d^2 + (2*c*(ArcCoth[c + d*x]^2/2 + ArcCoth[c +
d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])] - PolyLog[2, E^(-2*ArcCoth[c + d*x])]/2))/d^2))/((c + d*x)^2*(1 - (c +
d*x)^(-2))) + (b^2*e^2*(1 - (c + d*x)^2)*(ArcCoth[c + d*x]*(ArcCoth[c + d*x] - (c + d*x)*ArcCoth[c + d*x] + 2*
Log[1 - E^(-2*ArcCoth[c + d*x])]) - PolyLog[2, E^(-2*ArcCoth[c + d*x])]))/(d*(c + d*x)^2*(1 - (c + d*x)^(-2)))
- (b^2*f^2*(c + d*x)*Sqrt[1 - (c + d*x)^(-2)]*(1 - (c + d*x)^2)*((4*ArcCoth[c + d*x])/((c + d*x)*Sqrt[1 - (c
+ d*x)^(-2)]) + (3*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (12*c*ArcCoth[c + d*x]^2)/((c +
d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (9*c^2*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (-1 + 6*c*A
rcCoth[c + d*x] + 3*ArcCoth[c + d*x]^2 - 3*c^2*ArcCoth[c + d*x]^2)/Sqrt[1 - (c + d*x)^(-2)] + Cosh[3*ArcCoth[c
+ d*x]] - 6*c*ArcCoth[c + d*x]*Cosh[3*ArcCoth[c + d*x]] + ArcCoth[c + d*x]^2*Cosh[3*ArcCoth[c + d*x]] + 3*c^2
*ArcCoth[c + d*x]^2*Cosh[3*ArcCoth[c + d*x]] + (6*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])])/((c + d*x
)*Sqrt[1 - (c + d*x)^(-2)]) + (18*c^2*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])])/((c + d*x)*Sqrt[1 - (
c + d*x)^(-2)]) - (18*c*Log[(c + d*x)^(-1)])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (18*c*Log[1/Sqrt[1 - (c +
d*x)^(-2)]])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (4*(1 + 3*c^2)*PolyLog[2, E^(-2*ArcCoth[c + d*x])])/((c +
d*x)^3*(1 - (c + d*x)^(-2))^(3/2)) - ArcCoth[c + d*x]^2*Sinh[3*ArcCoth[c + d*x]] - 3*c^2*ArcCoth[c + d*x]^2*Si
nh[3*ArcCoth[c + d*x]] - 2*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])]*Sinh[3*ArcCoth[c + d*x]] - 6*c^2*
ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])]*Sinh[3*ArcCoth[c + d*x]] + 6*c*Log[(c + d*x)^(-1)]*Sinh[3*Ar
cCoth[c + d*x]] + 6*c*Log[1/Sqrt[1 - (c + d*x)^(-2)]]*Sinh[3*ArcCoth[c + d*x]]))/(12*d^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1410\) vs. \(2(360)=720\).

Time = 0.67 (sec) , antiderivative size = 1411, normalized size of antiderivative = 3.77

method	result	size
parts	\(\text {Expression too large to display}\)	\(1411\)
derivativedivides	\(\text {Expression too large to display}\)	\(1412\)
default	\(\text {Expression too large to display}\)	\(1412\)
risch	\(\text {Expression too large to display}\)	\(1687\)

[In]

int((f*x+e)^2*(a+b*arccoth(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/3*a^2*(f*x+e)^3/f+b^2/d*(1/3/d^2*f^2*arccoth(d*x+c)^2*(d*x+c)^3-1/d^2*f^2*arccoth(d*x+c)^2*(d*x+c)^2*c+1/d*f
*arccoth(d*x+c)^2*(d*x+c)^2*e+1/d^2*f^2*arccoth(d*x+c)^2*(d*x+c)*c^2-2/d*f*arccoth(d*x+c)^2*(d*x+c)*c*e+arccot
h(d*x+c)^2*(d*x+c)*e^2-1/3/d^2*f^2*arccoth(d*x+c)^2*c^3+1/d*f*arccoth(d*x+c)^2*c^2*e-arccoth(d*x+c)^2*c*e^2+1/
3*d/f*arccoth(d*x+c)^2*e^3+2/3/d^2/f*(1/2*arccoth(d*x+c)*f^3*(d*x+c)^2+1/2*arccoth(d*x+c)*ln(d*x+c-1)*f^3+1/2*
arccoth(d*x+c)*ln(d*x+c+1)*f^3-3/2*arccoth(d*x+c)*ln(d*x+c+1)*c^2*d*e*f^2+3/2*arccoth(d*x+c)*ln(d*x+c+1)*c*d^2
*e^2*f-3*arccoth(d*x+c)*ln(d*x+c+1)*c*d*e*f^2+3/2*arccoth(d*x+c)*ln(d*x+c+1)*d^2*e^2*f-3/2*arccoth(d*x+c)*ln(d
*x+c+1)*d*e*f^2+3/2*arccoth(d*x+c)*ln(d*x+c-1)*c^2*d*e*f^2-3/2*arccoth(d*x+c)*ln(d*x+c-1)*c*d^2*e^2*f-3*arccot
h(d*x+c)*ln(d*x+c-1)*c*d*e*f^2+3*arccoth(d*x+c)*d*e*f^2*(d*x+c)+3/2*arccoth(d*x+c)*ln(d*x+c-1)*d^2*e^2*f+3/2*a
rccoth(d*x+c)*ln(d*x+c-1)*d*e*f^2-1/2*arccoth(d*x+c)*ln(d*x+c-1)*c^3*f^3+1/2*arccoth(d*x+c)*ln(d*x+c-1)*d^3*e^
3+3/2*arccoth(d*x+c)*ln(d*x+c-1)*c^2*f^3-3/2*arccoth(d*x+c)*ln(d*x+c-1)*c*f^3+1/2*arccoth(d*x+c)*ln(d*x+c+1)*c
^3*f^3-1/2*arccoth(d*x+c)*ln(d*x+c+1)*d^3*e^3+3/2*arccoth(d*x+c)*ln(d*x+c+1)*c^2*f^3+3/2*arccoth(d*x+c)*ln(d*x
+c+1)*c*f^3-3*arccoth(d*x+c)*c*f^3*(d*x+c)+1/2*f^2*(f*(d*x+c)+1/2*(-6*c*f+6*d*e+f)*ln(d*x+c-1)-1/2*(6*c*f-6*d*
e+f)*ln(d*x+c+1))+1/2*(c^3*f^3-3*c^2*d*e*f^2+3*c*d^2*e^2*f-d^3*e^3+3*c^2*f^3-6*c*d*e*f^2+3*d^2*e^2*f+3*c*f^3-3
*d*e*f^2+f^3)*(-1/4*ln(d*x+c+1)^2+1/2*(ln(d*x+c+1)-ln(1/2*d*x+1/2*c+1/2))*ln(-1/2*d*x-1/2*c+1/2)-1/2*dilog(1/2
*d*x+1/2*c+1/2))+1/2*(-c^3*f^3+3*c^2*d*e*f^2-3*c*d^2*e^2*f+d^3*e^3+3*c^2*f^3-6*c*d*e*f^2+3*d^2*e^2*f-3*c*f^3+3
*d*e*f^2+f^3)*(1/4*ln(d*x+c-1)^2-1/2*dilog(1/2*d*x+1/2*c+1/2)-1/2*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2))))+2*a*b/d
*(1/3/d^2*f^2*arccoth(d*x+c)*(d*x+c)^3-1/d^2*f^2*arccoth(d*x+c)*(d*x+c)^2*c+1/d*f*arccoth(d*x+c)*(d*x+c)^2*e+1
/d^2*f^2*arccoth(d*x+c)*(d*x+c)*c^2-2/d*f*arccoth(d*x+c)*(d*x+c)*c*e+arccoth(d*x+c)*(d*x+c)*e^2-1/3/d^2*f^2*ar
ccoth(d*x+c)*c^3+1/d*f*arccoth(d*x+c)*c^2*e-arccoth(d*x+c)*c*e^2+1/3*d/f*arccoth(d*x+c)*e^3+1/3/d^2/f*(1/2*f^3
*(d*x+c)^2-3*c*f^3*(d*x+c)+3*d*e*f^2*(d*x+c)+1/2*(-c^3*f^3+3*c^2*d*e*f^2-3*c*d^2*e^2*f+d^3*e^3+3*c^2*f^3-6*c*d
*e*f^2+3*d^2*e^2*f-3*c*f^3+3*d*e*f^2+f^3)*ln(d*x+c-1)-1/2*(-c^3*f^3+3*c^2*d*e*f^2-3*c*d^2*e^2*f+d^3*e^3-3*c^2*
f^3+6*c*d*e*f^2-3*d^2*e^2*f-3*c*f^3+3*d*e*f^2-f^3)*ln(d*x+c+1)))

Fricas [F]

\[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

integrate((f*x+e)^2*(a+b*arccoth(d*x+c))^2,x, algorithm="fricas")

[Out]

integral(a^2*f^2*x^2 + 2*a^2*e*f*x + a^2*e^2 + (b^2*f^2*x^2 + 2*b^2*e*f*x + b^2*e^2)*arccoth(d*x + c)^2 + 2*(a
*b*f^2*x^2 + 2*a*b*e*f*x + a*b*e^2)*arccoth(d*x + c), x)

Sympy [F]

\[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )^{2}\, dx \]

[In]

integrate((f*x+e)**2*(a+b*acoth(d*x+c))**2,x)

[Out]

Integral((a + b*acoth(c + d*x))**2*(e + f*x)**2, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 791 vs. \(2 (350) = 700\).

Time = 0.42 (sec) , antiderivative size = 791, normalized size of antiderivative = 2.11 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {1}{3} \, a^{2} f^{2} x^{3} + a^{2} e f x^{2} + {\left (2 \, x^{2} \operatorname {arcoth}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} a b e f + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {arcoth}\left (d x + c\right ) + d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} + \frac {{\left (c^{3} + 3 \, c^{2} + 3 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{4}} - \frac {{\left (c^{3} - 3 \, c^{2} + 3 \, c - 1\right )} \log \left (d x + c - 1\right )}{d^{4}}\right )}\right )} a b f^{2} + a^{2} e^{2} x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {arcoth}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} a b e^{2}}{d} - \frac {{\left (3 \, d^{2} e^{2} - 6 \, c d e f + 3 \, c^{2} f^{2} + f^{2}\right )} {\left (\log \left (d x + c - 1\right ) \log \left (\frac {1}{2} \, d x + \frac {1}{2} \, c + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c + \frac {1}{2}\right )\right )} b^{2}}{3 \, d^{3}} - \frac {{\left (5 \, c^{2} f^{2} - 6 \, d e f - 6 \, {\left (d e f - f^{2}\right )} c + f^{2}\right )} b^{2} \log \left (d x + c + 1\right )}{6 \, d^{3}} + \frac {4 \, b^{2} d f^{2} x + {\left (b^{2} d^{3} f^{2} x^{3} + 3 \, b^{2} d^{3} e f x^{2} + 3 \, b^{2} d^{3} e^{2} x + {\left (c^{3} f^{2} + 3 \, d^{2} e^{2} - 3 \, {\left (d e f - f^{2}\right )} c^{2} - 3 \, d e f + 3 \, {\left (d^{2} e^{2} - 2 \, d e f + f^{2}\right )} c + f^{2}\right )} b^{2}\right )} \log \left (d x + c + 1\right )^{2} + {\left (b^{2} d^{3} f^{2} x^{3} + 3 \, b^{2} d^{3} e f x^{2} + 3 \, b^{2} d^{3} e^{2} x + {\left (c^{3} f^{2} - 3 \, d^{2} e^{2} - 3 \, {\left (d e f + f^{2}\right )} c^{2} - 3 \, d e f + 3 \, {\left (d^{2} e^{2} + 2 \, d e f + f^{2}\right )} c - f^{2}\right )} b^{2}\right )} \log \left (d x + c - 1\right )^{2} + 2 \, {\left (b^{2} d^{2} f^{2} x^{2} + 2 \, {\left (3 \, d^{2} e f - 2 \, c d f^{2}\right )} b^{2} x - {\left (b^{2} d^{3} f^{2} x^{3} + 3 \, b^{2} d^{3} e f x^{2} + 3 \, b^{2} d^{3} e^{2} x + {\left (c^{3} f^{2} - 3 \, d^{2} e^{2} - 3 \, {\left (d e f + f^{2}\right )} c^{2} - 3 \, d e f + 3 \, {\left (d^{2} e^{2} + 2 \, d e f + f^{2}\right )} c - f^{2}\right )} b^{2}\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right ) - 2 \, {\left (b^{2} d^{2} f^{2} x^{2} + 2 \, {\left (3 \, d^{2} e f - 2 \, c d f^{2}\right )} b^{2} x - {\left (5 \, c^{2} f^{2} + 6 \, d e f - 6 \, {\left (d e f + f^{2}\right )} c + f^{2}\right )} b^{2}\right )} \log \left (d x + c - 1\right )}{12 \, d^{3}} \]

[In]

integrate((f*x+e)^2*(a+b*arccoth(d*x+c))^2,x, algorithm="maxima")

[Out]

1/3*a^2*f^2*x^3 + a^2*e*f*x^2 + (2*x^2*arccoth(d*x + c) + d*(2*x/d^2 - (c^2 + 2*c + 1)*log(d*x + c + 1)/d^3 +
(c^2 - 2*c + 1)*log(d*x + c - 1)/d^3))*a*b*e*f + 1/3*(2*x^3*arccoth(d*x + c) + d*((d*x^2 - 4*c*x)/d^3 + (c^3 +
3*c^2 + 3*c + 1)*log(d*x + c + 1)/d^4 - (c^3 - 3*c^2 + 3*c - 1)*log(d*x + c - 1)/d^4))*a*b*f^2 + a^2*e^2*x +
(2*(d*x + c)*arccoth(d*x + c) + log(-(d*x + c)^2 + 1))*a*b*e^2/d - 1/3*(3*d^2*e^2 - 6*c*d*e*f + 3*c^2*f^2 + f^
2)*(log(d*x + c - 1)*log(1/2*d*x + 1/2*c + 1/2) + dilog(-1/2*d*x - 1/2*c + 1/2))*b^2/d^3 - 1/6*(5*c^2*f^2 - 6*
d*e*f - 6*(d*e*f - f^2)*c + f^2)*b^2*log(d*x + c + 1)/d^3 + 1/12*(4*b^2*d*f^2*x + (b^2*d^3*f^2*x^3 + 3*b^2*d^3
*e*f*x^2 + 3*b^2*d^3*e^2*x + (c^3*f^2 + 3*d^2*e^2 - 3*(d*e*f - f^2)*c^2 - 3*d*e*f + 3*(d^2*e^2 - 2*d*e*f + f^2
)*c + f^2)*b^2)*log(d*x + c + 1)^2 + (b^2*d^3*f^2*x^3 + 3*b^2*d^3*e*f*x^2 + 3*b^2*d^3*e^2*x + (c^3*f^2 - 3*d^2
*e^2 - 3*(d*e*f + f^2)*c^2 - 3*d*e*f + 3*(d^2*e^2 + 2*d*e*f + f^2)*c - f^2)*b^2)*log(d*x + c - 1)^2 + 2*(b^2*d
^2*f^2*x^2 + 2*(3*d^2*e*f - 2*c*d*f^2)*b^2*x - (b^2*d^3*f^2*x^3 + 3*b^2*d^3*e*f*x^2 + 3*b^2*d^3*e^2*x + (c^3*f
^2 - 3*d^2*e^2 - 3*(d*e*f + f^2)*c^2 - 3*d*e*f + 3*(d^2*e^2 + 2*d*e*f + f^2)*c - f^2)*b^2)*log(d*x + c - 1))*l
og(d*x + c + 1) - 2*(b^2*d^2*f^2*x^2 + 2*(3*d^2*e*f - 2*c*d*f^2)*b^2*x - (5*c^2*f^2 + 6*d*e*f - 6*(d*e*f + f^2
)*c + f^2)*b^2)*log(d*x + c - 1))/d^3

Giac [F]

\[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

integrate((f*x+e)^2*(a+b*arccoth(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((f*x + e)^2*(b*arccoth(d*x + c) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^2 \,d x \]

[In]

int((e + f*x)^2*(a + b*acoth(c + d*x))^2,x)

[Out]

int((e + f*x)^2*(a + b*acoth(c + d*x))^2, x)